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Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 23 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
The Garden of Knowledge as a Knowledge Manifold  A Conceptual Framework for Computer Supported Subjective Education
 CID17, TRITANAD9708, DEPARTMENT OF NUMERICAL ANALYSIS AND COMPUTING SCIENCE
, 1997
"... This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in te ..."
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Cited by 22 (14 self)
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This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols  procedures that are implemented on top of the participating subjective knowledgepatches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inversedirect limit duality of Category Theory.
The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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Cited by 5 (3 self)
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
Mathematical Masterpieces: Teaching with Original Sources
 In R. Calinger (Ed.), Vita Mathematica: Historical Research and Integration with Teaching
"... d down without human intervention. For instance, after reading Cayley's paper (see below) introducing abstract group theory, the student is much less bewildered upon seeing the axiomatic version so devoid of any motivation. Furthermore, Cayley makes the connection to the theory of algebraic equation ..."
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Cited by 4 (2 self)
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d down without human intervention. For instance, after reading Cayley's paper (see below) introducing abstract group theory, the student is much less bewildered upon seeing the axiomatic version so devoid of any motivation. Furthermore, Cayley makes the connection to the theory of algebraic equations, which a student might otherwise never become aware of. An additional feature of the method is that suddenly value judgments need to be made: there is good and bad mathematics, there are elegant proofs and clumsy ones, and of course plenty of mistakes and unsubstantiated assertions which need to be examined critically. Later follows the natural realization that new mathematics is being created even today, quite a surprise to many students. To achieve our aims we have selected mathematical masterpieces meeting the following criteria. First, sources must be original in the sense that new mathematics is captured in the words and notation of the inventor. Thus we assemble original works or En
Notes on Discrete Mathematics for Computer Scientists
, 2003
"... 1.2 Formal Languages.......................... 2 ..."
Cantor's Grundlagen and the Paradoxes of Set Theory
"... This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motiva ..."
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This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motivated my first attempts to understand proper classes and the realm of transfinite numbers. I read a version of the paper at the APA Central Division meeting in Chicago in May, 1998. I thank Howard Stein, who provided valuable criticisms of an earlier draft, ranging from the correction of spelling mistakes, through important historical remarks, to the correction of a mathematical mistake, and Patricia Blanchette, who commented on the paper at the APA meeting and raised two challenging points which have led to improvements in this final version
Consistency  What's Logic Got to Do with It?
, 1996
"... this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of cons ..."
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this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of consistency, borrowed primarily from the rigors of modern developments in logic, has prevented latter day twentieth century philosophers from producing philosophical systems of the type produced in earlier times.